Integrand size = 20, antiderivative size = 65 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=-\frac {d}{3 a x^3}-\frac {\sqrt {c} d \arctan \left (\frac {\sqrt {c} x^3}{\sqrt {a}}\right )}{3 a^{3/2}}+\frac {e \log (x)}{a}-\frac {e \log \left (a+c x^6\right )}{6 a} \] Output:
-1/3*d/a/x^3-1/3*c^(1/2)*d*arctan(c^(1/2)*x^3/a^(1/2))/a^(3/2)+e*ln(x)/a-1 /6*e*ln(c*x^6+a)/a
Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.91 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=\frac {-\frac {2 \sqrt {a} d}{x^3}+2 \sqrt {c} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )+2 \sqrt {c} d \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )-2 \sqrt {c} d \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+6 \sqrt {a} e \log (x)-\sqrt {a} e \log \left (a+c x^6\right )}{6 a^{3/2}} \] Input:
Integrate[(d + e*x^3)/(x^4*(a + c*x^6)),x]
Output:
((-2*Sqrt[a]*d)/x^3 + 2*Sqrt[c]*d*ArcTan[(c^(1/6)*x)/a^(1/6)] + 2*Sqrt[c]* d*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)] - 2*Sqrt[c]*d*ArcTan[Sqrt[3] + ( 2*c^(1/6)*x)/a^(1/6)] + 6*Sqrt[a]*e*Log[x] - Sqrt[a]*e*Log[a + c*x^6])/(6* a^(3/2))
Time = 0.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1803, 523, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx\) |
\(\Big \downarrow \) 1803 |
\(\displaystyle \frac {1}{3} \int \frac {e x^3+d}{x^6 \left (c x^6+a\right )}dx^3\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \frac {1}{3} \int \left (\frac {d}{a x^6}-\frac {c \left (e x^3+d\right )}{a \left (c x^6+a\right )}+\frac {e}{a x^3}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {\sqrt {c} d \arctan \left (\frac {\sqrt {c} x^3}{\sqrt {a}}\right )}{a^{3/2}}-\frac {e \log \left (a+c x^6\right )}{2 a}-\frac {d}{a x^3}+\frac {e \log \left (x^3\right )}{a}\right )\) |
Input:
Int[(d + e*x^3)/(x^4*(a + c*x^6)),x]
Output:
(-(d/(a*x^3)) - (Sqrt[c]*d*ArcTan[(Sqrt[c]*x^3)/Sqrt[a]])/a^(3/2) + (e*Log [x^3])/a - (e*Log[a + c*x^6])/(2*a))/3
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x )^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {d}{3 a \,x^{3}}+\frac {e \ln \left (x \right )}{a}-\frac {c \left (\frac {e \ln \left (c \,x^{6}+a \right )}{2 c}+\frac {d \arctan \left (\frac {c \,x^{3}}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{3 a}\) | \(57\) |
risch | \(-\frac {d}{3 a \,x^{3}}+\frac {e \ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{2} a^{3}+2 \textit {\_Z} e \,a^{2}+a \,e^{2}+c \,d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-7 a^{3} \textit {\_R}^{2}-7 a^{2} e \textit {\_R} -6 c \,d^{2}\right ) x^{3}-a^{2} d \textit {\_R} +6 a d e \right )\right )}{6}\) | \(91\) |
Input:
int((e*x^3+d)/x^4/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-1/3*d/a/x^3+e*ln(x)/a-1/3*c/a*(1/2*e*ln(c*x^6+a)/c+d/(a*c)^(1/2)*arctan(c *x^3/(a*c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.15 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=\left [\frac {d x^{3} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{6} - 2 \, a x^{3} \sqrt {-\frac {c}{a}} - a}{c x^{6} + a}\right ) - e x^{3} \log \left (c x^{6} + a\right ) + 6 \, e x^{3} \log \left (x\right ) - 2 \, d}{6 \, a x^{3}}, -\frac {2 \, d x^{3} \sqrt {\frac {c}{a}} \arctan \left (x^{3} \sqrt {\frac {c}{a}}\right ) + e x^{3} \log \left (c x^{6} + a\right ) - 6 \, e x^{3} \log \left (x\right ) + 2 \, d}{6 \, a x^{3}}\right ] \] Input:
integrate((e*x^3+d)/x^4/(c*x^6+a),x, algorithm="fricas")
Output:
[1/6*(d*x^3*sqrt(-c/a)*log((c*x^6 - 2*a*x^3*sqrt(-c/a) - a)/(c*x^6 + a)) - e*x^3*log(c*x^6 + a) + 6*e*x^3*log(x) - 2*d)/(a*x^3), -1/6*(2*d*x^3*sqrt( c/a)*arctan(x^3*sqrt(c/a)) + e*x^3*log(c*x^6 + a) - 6*e*x^3*log(x) + 2*d)/ (a*x^3)]
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (56) = 112\).
Time = 1.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.20 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=\left (- \frac {e}{6 a} - \frac {d \sqrt {- a^{3} c}}{6 a^{3}}\right ) \log {\left (x^{3} + \frac {- 6 a^{2} \left (- \frac {e}{6 a} - \frac {d \sqrt {- a^{3} c}}{6 a^{3}}\right ) - a e}{c d} \right )} + \left (- \frac {e}{6 a} + \frac {d \sqrt {- a^{3} c}}{6 a^{3}}\right ) \log {\left (x^{3} + \frac {- 6 a^{2} \left (- \frac {e}{6 a} + \frac {d \sqrt {- a^{3} c}}{6 a^{3}}\right ) - a e}{c d} \right )} - \frac {d}{3 a x^{3}} + \frac {e \log {\left (x \right )}}{a} \] Input:
integrate((e*x**3+d)/x**4/(c*x**6+a),x)
Output:
(-e/(6*a) - d*sqrt(-a**3*c)/(6*a**3))*log(x**3 + (-6*a**2*(-e/(6*a) - d*sq rt(-a**3*c)/(6*a**3)) - a*e)/(c*d)) + (-e/(6*a) + d*sqrt(-a**3*c)/(6*a**3) )*log(x**3 + (-6*a**2*(-e/(6*a) + d*sqrt(-a**3*c)/(6*a**3)) - a*e)/(c*d)) - d/(3*a*x**3) + e*log(x)/a
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=-\frac {c d \arctan \left (\frac {c x^{3}}{\sqrt {a c}}\right )}{3 \, \sqrt {a c} a} - \frac {e \log \left (c x^{6} + a\right )}{6 \, a} + \frac {e \log \left (x^{3}\right )}{3 \, a} - \frac {d}{3 \, a x^{3}} \] Input:
integrate((e*x^3+d)/x^4/(c*x^6+a),x, algorithm="maxima")
Output:
-1/3*c*d*arctan(c*x^3/sqrt(a*c))/(sqrt(a*c)*a) - 1/6*e*log(c*x^6 + a)/a + 1/3*e*log(x^3)/a - 1/3*d/(a*x^3)
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=-\frac {c d \arctan \left (\frac {c x^{3}}{\sqrt {a c}}\right )}{3 \, \sqrt {a c} a} - \frac {e \log \left (c x^{6} + a\right )}{6 \, a} + \frac {e \log \left ({\left | x \right |}\right )}{a} - \frac {e x^{3} + d}{3 \, a x^{3}} \] Input:
integrate((e*x^3+d)/x^4/(c*x^6+a),x, algorithm="giac")
Output:
-1/3*c*d*arctan(c*x^3/sqrt(a*c))/(sqrt(a*c)*a) - 1/6*e*log(c*x^6 + a)/a + e*log(abs(x))/a - 1/3*(e*x^3 + d)/(a*x^3)
Time = 20.99 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=\frac {e\,\ln \left (x\right )}{a}-\frac {d}{3\,a\,x^3}-\frac {e\,\ln \left (c\,x^6+a\right )}{6\,a}-\frac {\sqrt {c}\,d\,\mathrm {atan}\left (\frac {c^{3/2}\,d^2\,x^3}{\sqrt {a}\,\left (c\,d^2+49\,a\,e^2\right )}+\frac {49\,\sqrt {a}\,\sqrt {c}\,e^2\,x^3}{c\,d^2+49\,a\,e^2}\right )}{3\,a^{3/2}} \] Input:
int((d + e*x^3)/(x^4*(a + c*x^6)),x)
Output:
(e*log(x))/a - d/(3*a*x^3) - (e*log(a + c*x^6))/(6*a) - (c^(1/2)*d*atan((c ^(3/2)*d^2*x^3)/(a^(1/2)*(49*a*e^2 + c*d^2)) + (49*a^(1/2)*c^(1/2)*e^2*x^3 )/(49*a*e^2 + c*d^2)))/(3*a^(3/2))
Time = 0.19 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.40 \[ \int \frac {d+e x^3}{x^4 \left (a+c x^6\right )} \, dx=\frac {2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d \,x^{3}-2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 c^{\frac {1}{3}} x}{c^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) d \,x^{3}+2 c^{\frac {7}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {c^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) d \,x^{3}-c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e \,x^{3}-c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e \,x^{3}-c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e \,x^{3}+6 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (x \right ) e \,x^{3}-2 c^{\frac {2}{3}} a^{\frac {2}{3}} d}{6 c^{\frac {2}{3}} a^{\frac {5}{3}} x^{3}} \] Input:
int((e*x^3+d)/x^4/(c*x^6+a),x)
Output:
(2*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqrt(3) - 2*c**(1/3)*x)/(c**( 1/6)*a**(1/6)))*c*d*x**3 - 2*c**(1/6)*a**(1/6)*atan((c**(1/6)*a**(1/6)*sqr t(3) + 2*c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d*x**3 + 2*c**(1/6)*a**(1/6)*a tan((c**(1/3)*x)/(c**(1/6)*a**(1/6)))*c*d*x**3 - c**(2/3)*a**(2/3)*log(a** (1/3) + c**(1/3)*x**2)*e*x**3 - c**(2/3)*a**(2/3)*log( - c**(1/6)*a**(1/6) *sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*e*x**3 - c**(2/3)*a**(2/3)*log(c**( 1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + c**(1/3)*x**2)*e*x**3 + 6*c**(2/3)*a* *(2/3)*log(x)*e*x**3 - 2*c**(2/3)*a**(2/3)*d)/(6*c**(2/3)*a**(2/3)*a*x**3)